DOCSLIB.ORG

Listing Acyclic Orientations of Graphs with Single and Multiple Sources Alessio Conte, Roberto Grossi, Andrea Marino, Romeo Rizzi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Listing Acyclic Orientations of Graphs with Single and Multiple Sources Alessio Conte, Roberto Grossi, Andrea Marino, Romeo Rizzi Listing Acyclic Orientations of Graphs with Single and Multiple Sources Alessio Conte, Roberto Grossi, Andrea Marino, Romeo Rizzi To cite this version: Alessio Conte, Roberto Grossi, Andrea Marino, Romeo Rizzi. Listing Acyclic Orientations of Graphs with Single and Multiple Sources. LATIN 2016: Theoretical Informatics - 12th Latin American Sym- posium, Apr 2016, Ensenada, Mexico. pp.319-333, 10.1007/978-3-662-49529-2_24. hal-01388470 HAL Id: hal-01388470 https://hal.inria.fr/hal-01388470 Submitted on 30 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Listing Acyclic Orientations of Graphs with Single and Multiple Sources Alessio Conte1, Roberto Grossi1, Andrea Marino1, and Romeo Rizzi2 1 Università di Pisa and Erable, Inria, conte,grossi,[email protected]? 2 Università di Verona, [email protected] Abstract. We study enumeration problems for the acyclic orientations of an undirected graph with n nodes and m edges, where each edge must be assigned a direction so that the resulting directed graph is acyclic. When the acyclic orientations have single or multiple sources specified as input along with the graph, our algorithm is the first one to provide guaranteed bounds, giving new bounds with a delay of O(m · n) time per solution and O(n2) working space. When no sources are specified, our algorithm improves over previous work by reducing the delay to O(m), and is the first one with linear delay. 1 Introduction Acyclic orientations of graphs are related to several basic problems in graph −! theory. An orientation of an undirected graph G is the directed graph G whose −! arcs are obtained assigning a direction to each edge in G. The orientation G is −! acyclic when it does not contain cycles, and a node s is a source in G if it has indegree zero. For instance, consider the graphs in Figure 1. The directed graph in (b) is an acyclic orientation for the undirected graph in (a). In particular, since the orientation has only one source (v9), it is called a single source acyclic orientation. Starting from the observation that each acyclic orientation corresponds to a partial order for the underlying graph, Iriarte [7] investigates which orientations maximize the number of linear extensions of the corresponding poset. Alon and Tarsi [1] look for special orientations to give bounds on the size of the maximum independent set or the chromatic number. Gallai, Roy, and Vitaver indepen- dently describe a well-known result stating that every orientation of a graph with chromatic number k contains a simple directed path with k vertices [6, 12, 17]. There are further problems that can be addressed by looking at acyclic orientations. For instance, Benson et al. [4] show that there exists a bijection between the set of the so-called superstable configurations of a graph and the set of its acyclic orientations with a unique source. ? This work has been partially supported by the Italian Ministry of Education, Uni- versity, and Research (MIUR) under PRIN 2012C4E3KT national research project AMANDA — Algorithmics for MAssive and Networked DAta. v6 v6 v3 v9 v3 v9 v8 v8 v2 v2 v5 v7 v5 v7 v4 v4 v1 v1 (a) (b) Fig. 1. An undirected connected graph without self-loops (a) and one of its acyclic orientations (b). Counting how many acyclic orientations can be found in a graph is a funda- mental problem in combinatorics, dating back to the 70s or earlier [16]. Linial [10] proves that this problem is #P-complete. Stanley [15] shows how the number of acyclic orientations can be computed by using the chromatic polynomial (a special case of Tutte’s polynomial). Another approach that concerns the number of acyclic orientations is the acyclic orientation game. Alon and Tuza [2] inquire about the amount of oriented edges needed to define a unique orientation of G, and find this number to be almost surely Θ(jV j log jV j) in Erdős–Rényi random graphs. Pikhurko [11] shows that the number of these edges in the worst case is 1 2 no greater than ( 4 + o(1))jV j for general graphs. Problems addressed. Our paper investigates new algorithms for enumerat- ing patterns for this interesting problem in graph theory, given an undirected connected graph G(V; E) with n nodes and m edges. single source acyclic orientations ( ): Given a node s 2 V , enumerate −! ssao all the acyclic orientations G of G, such that s is the only source. single source acyclic orientations (weak ): Given a set of nodes S ⊆ −!ssao V , enumerate all the acyclic orientations G of G such that there is exactly one source x and x 2 S. multiple source acyclic orientations (strong ): Given a set of nodes −!msao S ⊆ V , enumerate all the acyclic orientations G of G such that all the nodes in S are the only sources.3 multiple source acyclic orientations (weak ): Given a set of nodes msao−! S ⊆ V , enumerate all the acyclic orientations G of G such that if x is a source then x 2 S.4 −! acyclic orientations (ao): Enumerate all the acyclic orientations G of G. We will show that these problems can be reduced to ssao, with a one-to-one correspondence between their solutions. Many other variants with constraints on the number or choice of sources can be reduced to ssao as well: the ones 3 These orientations are possible if and only if S is an independent set. 4 Not all nodes in S must be sources, but there cannot be sources in V n S. 2 we present are some of the most representative ones. We analyze the cost of an enumeration algorithm for ssao, weak ssao, strong msao, weak msao, and ao in terms of its delay cost, which is a well-known measure of performance for enumeration algorithms corresponding to the worst-case time between any two consecutively enumerated solutions (e.g. [8]). We are interested in algorithms with guaranteed delay and space. Previous work. We are not aware of any provably good bounds for problems ssao, weak ssao, strong msao and weak msao. Johnson’s backtracking algo- rithm [9] for ssao has been presented over 30 years ago to solve problems on network reliability. However its complexity is not given and is hard to estimate, as it is based on a backtracking approach with dead ends. In his paper presenting an algorithm for ao, Squire [14] writes that he has been unable to efficiently implement Johnson’s approach because of its dead ends. Squire’s algorithm for ao uses Gray codes and has an amortized cost of O(n) per solution, but its delay can be O(n3) time for a solution. The algorithm by Barbosa and Szwarcfiter [3] solves ao with an amortized time complexity of O(n + m) per solution, delay O(n · m). The algorithm builds the oriented graph incrementally by iteratively adding the nodes to an empty directed graph. It is worth observing that by replacing each edge with a double arc and applying any algorithm for maximal feedback arc set enumeration, one can obtain all the acyclic orientations of G. State of the art approaches for the latter problem guarantee a delay Ω(n3) as shown by Schwikowski and Speckenmeyer [13]. All the techniques above for ao, including the one in [14], do not to extend smoothly to ssao, weak ssao, strong msao, and weak msao. In a previous work [5], we studied the related problem of enumerating the cyclic orientations of an undirected graph, but the proposed techniques cannot be reused for the problems in this paper. Our results. Our contribution is the design of the first enumeration algorithms with guaranteed bounds for ssao, weak ssao, strong msao, and weak msao: the complexity is O(m · n) delay per solution using O(n2) space. For ao, we also show how to obtain O(m) delay, improving the delay of [3, 13, 14], but we do not improve the amortized cost of O(n) in [14]. We therefore focus on ssao in the first part of the paper, and then show an optimization that holds for the case of ao. We guarantee that, at any given partial solution, the extensions of the partial solution will enumerate new acyclic orientations. To this aim, we solve several non-trivial issues. – We use a recursive approach where each call surely leads to a solution. For ssao this is achieved also by using a suitable ordering of the nodes. – We quickly identify the next recursion calls within the claimed time delay. – We do a careful analysis of the recursion tree, and show how to check effi- ciently for node reachability during recursion. – In the case of ao we exploit the fact that the recursion tree does not contain unary nodes. 3 The paper is organized as follows. We give the necessary definitions and terminology in Section 2. We then discuss how to solve ssao in Section 3 and further reduce the delay for ao in Section 4. In Section 5 we show how weak ssao, strong msao, weak msao, and ao reduce to ssao. We draw some conclusions in Section 6. 2 Preliminaries Given an undirected graph G(V; E) with n nodes and m edges, an orientation of −! −! −! G is the directed graph G(V; E ) where for any pair fu; vg 2 E either (u; v) 2 E −! −! −! or (v; u) 2 E .
Load more

Recommended publications
  • I?'!'-Comparability Graphs
    Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors).
    [Show full text]
  • Graph Orientations and Linear Extensions
    GRAPH ORIENTATIONS AND LINEAR EXTENSIONS. BENJAMIN IRIARTE Abstract. Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a par- tial order on the vertices of our graph and, therefore, we can count the number of linear extensions of these posets. We want to know which choice of orien- tation maximizes the number of linear extensions of the corresponding poset, and this problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple graphs will be studied, including the case of random graphs; this will culminate in 1) new bounds for the volume of the stable polytope and 2) strong concentration results for our main statistic and for the graph entropy, which hold true a:s: for random graphs. We will then argue that our problem springs up naturally in the theory of graphical arrangements and graphical zonotopes. 1. Introduction. Linear extensions of partially ordered sets have been the object of much attention and their uses and applications remain increasing. Their number is a fundamental statistic of posets, and they relate to ever-recurring problems in computer science due to their role in sorting problems. Still, many fundamental questions about linear extensions are unsolved, including the well-known 1=3-2=3 Conjecture. Efficiently enumerating linear extensions of certain posets is difficult, and the general problem has been found to be ]P-complete in Brightwell and Winkler (1991). Directed acyclic graphs, and similarly, acyclic orientations of simple undirected graphs, are closely related to posets, and their problem-modeling values in several disciplines, including the biological sciences, needs no introduction.
    [Show full text]
  • Dessin De Triangulations: Algorithmes, Combinatoire, Et Analyse
    Dessin de triangulations: algorithmes, combinatoire, et analyse Eric´ Fusy Projet ALGO, INRIA Rocquencourt et LIX, Ecole´ Polytechnique – p.1/53 Motivations Display of large structures on a planar surface – p.2/53 Planar maps • A planar map is obtained by embedding a planar graph in the plane without edge crossings. • A planar map is defined up to continuous deformation Planar map The same planar map Quadrangulation Triangulation – p.3/53 Planar maps • Planar maps are combinatorial objects • They can be encoded without dealing with coordinates 2 1 Planar map Choice of labels 6 7 4 3 5 Encoding: to each vertex is associated the (cyclic) list of its neighbours in clockwise order 1: (2, 4) 2: (1, 7, 6) 3: (4, 6, 5) 4: (1, 3) 5: (3, 7) 6: (3, 2, 7) 7: (2, 5, 6) – p.4/53 Combinatorics of maps Triangulations Quadrangulations Tetravalent 3 spanning trees 2 spanning trees eulerian orientation = 2(4n−3)! = 2(3n−3)! = 2·3n(2n)! |Tn| n!(3n−2)! |Qn| n!(2n−2)! |En| n!(n+2)! ⇒ Tutte, Schaeffer, Schnyder, De Fraysseix et al... – p.5/53 A particular family of triangulations • We consider triangulations of the 4-gon (the outer face is a quadrangle) • Each 3-cycle delimits a face (irreducibility) Forbidden Irreducible – p.6/53 Transversal structures We define a transversal structure using local conditions Inner edges are colored blue or red and oriented: Inner vertex ⇒ Border vertices ⇒ Example: ⇒ – p.7/53 Link with bipolar orientations bipolar orientation = acyclic orientation with a unique minimum and a unique maximum The blue (resp.
    [Show full text]
  • Acyclic Orientations of Graphsଁ Richard P
    Discrete Mathematics 306 (2006) 905–909 www.elsevier.com/locate/disc Acyclic orientations of graphsଁ Richard P. Stanley Department of Mathematics, University of California, Berkeley, Calif. 94720, USA Abstract Let G be a finite graph with p vertices and its chromatic polynomial. A combinatorial interpretation is given to the positive p p integer (−1) (−), where is a positive integer, in terms of acyclic orientations of G. In particular, (−1) (−1) is the number of acyclic orientations of G. An application is given to the enumeration of labeled acyclic digraphs. An algebra of full binomial type, in the sense of Doubilet–Rota–Stanley, is constructed which yields the generating functions which occur in the above context. © 1973 Published by Elsevier B.V. 1. The chromatic polynomial with negative arguments Let G be a finite graph, which we assume to be without loops or multiple edges. Let V = V (G) denote the set of vertices of G and X = X(G) the set of edges. An edge e ∈ X is thought of as an unordered pair {u, v} of two distinct vertices. The integers p and q denote the cardinalities of V and X, respectively. An orientation of G is an assignment of a direction to each edge {u, v}, denoted by u → v or v → u, as the case may be. An orientation of G is said to be acyclic if it has no directed cycles. Let () = (G, ) denote the chromatic polynomial of G evaluated at ∈ C.If is a non-negative integer, then () has the following rather unorthodox interpretation. Proposition 1.1.
    [Show full text]
  • Parameterized Mixed Graph Coloring
    Parameterized mixed graph coloring Downloaded from: https://research.chalmers.se, 2021-09-26 10:22 UTC Citation for the original published paper (version of record): Damaschke, P. (2019) Parameterized mixed graph coloring Journal of Combinatorial Optimization, 38(2): 362-374 http://dx.doi.org/10.1007/s10878-019-00388-z N.B. When citing this work, cite the original published paper. research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology. It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004. research.chalmers.se is administrated and maintained by Chalmers Library (article starts on next page) Journal of Combinatorial Optimization (2019) 38:362–374 https://doi.org/10.1007/s10878-019-00388-z Parameterized Mixed Graph Coloring Peter Damaschke1 Published online: 7 February 2019 © The Author(s) 2019 Abstract Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NP- complete. Furthermore we point out a new polynomial case where the edges form a clique.
    [Show full text]
  • The Number of Orientations Having No Fixed Tournament
    The number of orientations having no fixed tournament Noga Alon ∗ Raphael Yuster y Abstract Let T be a fixed tournament on k vertices. Let D(n; T ) denote the maximum number of tk 1(n) orientations of an n-vertex graph that have no copy of T . We prove that D(n; T ) = 2 − for all sufficiently (very) large n, where tk 1(n) is the maximum possible number of edges of a − graph on n vertices with no Kk, (determined by Tur´an'sTheorem). The proof is based on a directed version of Szemer´edi'sregularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that n2=4 avoid the use of the regularity lemma and therefore show that in these cases D(n; T ) = 2b c already holds for (relatively) small values of n. 1 Introduction All graphs considered here are finite and simple. For standard terminology on undirected and directed graphs the reader is referred to [4]. Let T be some fixed tournament. An orientation of an undirected graph G = (V; E) is called T -free if it does not contain T as a subgraph. Let D(G; T ) denote the number of orientations of G that are T -free. Let D(n; T ) denote the maximum possible value of D(G; T ) where G is an n-vertex graph. In this paper we determine D(n; T ) precisely for every fixed tournament T and all sufficiently large n.
    [Show full text]
  • The Connectivity of Acyclic Orientation Graphs Carla D
    Discrete Mathematics 184 (1998) 281{287 Note The connectivity of acyclic orientation graphs Carla D. Savagea;∗;1, Cun-Quan Zhangb;2 a Department of Computer Science, North Carolina State University, Raleigh, NC 27695-8206, USA b Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA Received 27 July 1995; received in revised form 5 May 1997; accepted 19 May 1997 Abstract The acyclic orientation graph, AO(G), of an undirected graph, G, is the graph whose vertices are the acyclic orientations of G and whose edges are the pairs of orientations di ering only by the reversal of one edge. Edelman (1984) has observed that it follows from results on polytopes that when G is simple, the connectivity of AO(G) is at least n − c, where n is the number of vertices and c is the number of components of G. In this paper we give a simple graph-theoretic proof of this fact. Our proof uses a result of independent interest. We establish that if H is a triangle-free graph with minimum degree at least k, and the graph obtained by contracting the edges of a matching in H is k-connected, then H is k-connected. The connectivity bound on AO(G) is tight for various graphs including Kn, Kp; q, and trees. Applications and extensions are discussed, as well as the connection with polytopes. c 1998 Elsevier Science B.V. All rights reserved 1. Introduction For an undirected graph G and an acyclic orientation D of G, an edge e of D is called dependent if reversing the direction of e creates a directed cycle in D; otherwise, e is independent [18].
    [Show full text]
  • Approximating the Longest Path Length of a Stochastic DAG by a Normal Distribution in Linear Time ∗ Ei Ando A, , Toshio Nakata B, Masafumi Yamashita A,C
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Discrete Algorithms 7 (2009) 420–438 Contents lists available at ScienceDirect Journal of Discrete Algorithms www.elsevier.com/locate/jda Approximating the longest path length of a stochastic DAG by a normal distribution in linear time ∗ Ei Ando a, , Toshio Nakata b, Masafumi Yamashita a,c a Dept. Computer Sci. and Communication Eng., Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan b Dept. of Mathematics, Fukuoka University of Education, Akama-Bunkyomachi, Munakata, Fukuoka, 811-4192, Japan c Institute of Systems, Information Technologies and Nanotechnologies, Fukuoka SRP Center Building 7F 2-1-22, Momochihama, Sawara-ku, Fukuoka, 814-0001, Japan article info abstract Article history: This paper presents a linear time algorithm for approximating, in the sense below, the Received 3 September 2007 longest path length of a given directed acyclic graph (DAG), where each edge length is Received in revised form 7 January 2009 given as a normally distributed random variable. Let F (x) be the distribution function of Accepted 12 January 2009 the longest path length of the DAG. Our algorithm computes the mean and the variance Available online 21 January 2009 of a normal distribution whose distribution function F˜ (x) satisfies F˜ (x) F (x) as long as Keywords: F (x) a,givenaconstanta (1/2 a < 1). In other words, it computes an upper bound ˜ −1 Directed acyclic graph 1 − F (x) on the tail probability 1 − F (x),providedx F (a). To evaluate the accuracy ˜ Longest path problem of the approximation of F (x) by F (x), we first conduct two experiments using a standard Stochastic edge weight benchmark set ITC’99 of logical circuits, since a typical application of the algorithm is the Normal distribution delay analysis of logical circuits.
    [Show full text]
  • V. Pilaud Algebraic and Geometric Combinatorics of Lattice Quotients of the Weak Order
    Journ´ees du GT Combinatoire Alg´ebrique 26{27 juin 2018 V. PILAUD (CNRS & LIX) ALGEBRAIC AND GEOMETRIC COMBINATORICS OF LATTICE QUOTIENTS OF THE WEAK ORDER LATTICE QUOTIENTS AND GEOMETRY FANS & POLYTOPES Ziegler, Lectures on polytopes ('95) Matouˇsek, Lectures on Discrete Geometry ('02) FANS polyhedral cone = positive span of a finite set of Rd = intersection of finitely many linear half-spaces fan = collection of polyhedral cones closed by faces and where any two cones intersect along a face simplicial fan = maximal cones generated by d rays POLYTOPES polytope = convex hull of a finite set of Rd = bounded intersection of finitely many affine half-spaces face = intersection with a supporting hyperplane face lattice = all the faces with their inclusion relations simple polytope = facets in general position = each vertex incident to d facets SIMPLICIAL COMPLEXES, FANS, AND POLYTOPES P polytope, F face of P normal cone of F = positive span of the outer normal vectors of the facets containing F normal fan of P = f normal cone of F j F face of P g simple polytope =) simplicial fan =) simplicial complex WEAK ORDER AND PERMUTAHEDRON WEAK ORDER inversions of σ 2 Sn = pair (σi; σj) such that i < j and σi > σj weak order = permutations of Sn ordered by inclusion of inversion sets 4321 3421 4231 4312 3241 2431 3412 4213 4132 3214 2341 3142 2413 4123 1432 2314 3124 2143 1342 1423 2134 1324 1243 1234 PERMUTAHEDRON Permutahedron Perm(n) = conv f(τ(1); : : : ; τ(n)) j τ 2 Sng \ n X jJj + 1 = H \ x 2 R xj ≥ 2 4321 ?6=J([n] j2J 4312 X = 11+ [e ; e ] 3421
    [Show full text]
  • Gray Code Results for Acyclic Orientations 1 Introduction
    Gray Code Results for Acyclic Orientations Carla D. Savage,∗ Matthew B. Squire,† and Douglas B. West‡ Abstract Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of the acyclic orientations of G. We prove that for certain graphs G, AO(G) is hamiltonian, and give explicit constructions of hamilton cycles or paths. This work includes Gray codes for listing the acyclic orientations of trees, complete graphs, odd cycles, chordal graphs, odd ladder graphs, and odd wheel graphs. We also give examples of graphs whose acyclic orientation graph is not hamiltonian. We show that the acyclic orientations of even cycles, some complete bipartite graphs, even ladder graphs, and even wheel graphs cannot be listed by the defined Gray code. 1 Introduction A Gray code for a set S is a listing of the elements of S such that successive elements differ by a small amount. The classic example has the set S being the binary strings of length n and the small amount being the complementation of a single bit. Gray codes exist for a wide variety of combinatorial sets, including permutations [Tr62, Jo63], binary trees [LRR], integer partitions [Sa89, RSW], and linear extensions of some posets [Ru92, PR91]. In this paper, we study Gray codes for acyclic orientations of graphs. Let G = (V (G),E(G)) be an undirected graph.
    [Show full text]
  • Extremal Oriented Graphs and Erd˝Os-Hajnal Conjecture
    EXTREMAL ORIENTED GRAPHS AND ERDOS-HAJNAL} CONJECTURE by Payam Valadkhan B.Sc., Sharif University of Technology, 2007 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the School of Computing Science c Payam Valadkhan 2009 SIMON FRASER UNIVERSITY Summer 2009 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Payam Valaclkhan Degree: N{aster o[ Science Title of Thesis: Extrernal OrierrteclGraphs and Erd6s-Hajnal Conjecture Examining Committee: Dr. RarrreshKrisirnamurti Cirair Dr. Gd,bor Tardos, Professor',Compnting Science Sirnori Fra,serUniversity Seriior Supervisor Dr. Arrrlrei Bulat,ov. Associate Professor, Ooniputing Scierrce SirrronF'raser L rriversitv Superrvisor Dr. LaclislarvStacho. Associate Professor. N,{at}renriit- ics Simorr Fraser University SFU Exanriner Date Approved: /ll,tnp vl tll- 4'1-lq| ./ ilu ii^rl-)r,C,') ,/ !,t Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the “Institutional Repository” link of the SFU Library website <www.lib.sfu.ca> at: <http://ir.lib.sfu.ca/handle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work.
    [Show full text]
  • Thesis a Parametric Classification of Directed Acyclic Graphs Submitted by Mmanu Chaturvedi Department of Computer Science in Pa
    Thesis A Parametric Classification of Directed Acyclic Graphs Submitted by Mmanu Chaturvedi Department of Computer Science In partial fulfillment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Summer 2017 Master's Committee: Advisor: Ross M. McConnell Michael J. Kirby Sanjay V. Rajopadhye Iuliana Oprea Copyright by Mmanu Chaturvedi 2017 All Rights Reserved Abstract A Parametric Classification of Directed Acyclic Graphs We consider four NP-hard optimization problems on directed acyclic graphs (DAGs), namely, max clique, min coloring, max independent set and min clique cover. It is well- known that these four problems can be solved in polynomial time on transitive DAGs. It is also known that there can be no polynomial O(n1−)-approximation algorithms for these problems on the general class of DAGs unless P = NP . We propose a new parameter, β, as a measure of departure from transitivity for DAGs. We define β to be the number of vertices in a longest path in a DAG such that there is no edge from the first to the last vertex of the path, and 2 if the graph is transitive. Different values of β define a hierarchy of classes of DAGs, starting with the class of transitive DAGs. We give a polynomial time algorithm for finding a max clique when β is bounded by a fixed constant. The algorithm is exponential in β, but we also give a polynomial β-approximation algorithm. We prove that the other three decision problems are NP -hard even for β ≥ 4 and give polynomial algorithms with approximation bounds of β or better in each case.
    [Show full text]